Suppose (C, t) is a metric site. Denote by t^: C^{^} ® Top the Kan extensions of t on C^{^}. We obtain a presite (C^{^}, t^). Denote by (C^~, t^~) the subpresite of C^{^} with the induced metric functor t^~ = t^|_C~. We shall see that (C^{^}, t^) and (C^~, t^~) are metric sites. 

Remark 2.3.1. If C is strict, then C Í C^~ and t^~ is the Kan extension of t on C^~. 

Proposition 2.3.2. (C^{^}, t^) and (C^~, t^~) are everywhere active. 

Proof. If U is a subset of the space |A| of a presheaf A Î C^{^}. Then U determines a subfunctor U of A given by U(Z) = {f Î A(Z) ½ |f(Z)| Í U} for each Z Î C. Clearly the canonical morphism h: U ® A is active in C^{^}. Thus C^{^} is everywhere active. If A Î C^~ is a sheaf on C the subfunctor U of A is also a sheaf on C. Then h: U ® A is a active morphism in C^~. Thus C^~ is everywhere active. 

Theorem 2.3.3. (C^{^}, t^) and (C^~, t^~) are effective metric sites. 

Proof. Since C is exact and separable, C^{^} is exact and separable by (1.4.13) and (1.4.17). Also C^{^} is active (2.3.2). Thus C is effective, hence a metric site. Next we assume C is strict. Then C^~ is effective by the proof of (2.3.2). Since fibre products exist C^~ and the inclusion functor C^~ ® C^{^} preserves fibre products (2.2.7 or 2.2.8), C^~ is separable by (1.3.2). Thus C^~ is a metric site. (If C is strict then this also follows from (1.4.13) and (1.4.17) in view of (2.3.1).) 

Theorem 2.3.4. Suppose C is a strict metric site. Then C^~ is a Cauchy-complete metric site containing C as a subsite. 

Proof. We prove that C is strict by verifying the definition (2.1.2) of strict metric sites for two sheaves A, B Î C^~ and an open cover {U_i} of |A|. Suppose f, g Î hom (A, B) and f ¹ g. We can find Z Î C and h Î A(Z) = hom (Z, A) such that fh ¹ gh. Let {V_j} be an open effective cover of |Z| such that each h(V_j) is contained in some U_i. Since B is a sheaf on C, and Z Î C, we can find an open subset V_k such that the restrictions of fh and gh to V_k are different. Suppose h(V_k) is contained U_k. Then the restrictions of f and g to U_k are different. 
Now suppose for any i we have an f_i Î hom (U_i, B) such that the restrictions of f_i and f_j to U_i Ç U_j are the same. For any object Z Î C and h Î hom (Z, A), let {V_j} be an open effective cover of |Z| as above. Denote by h_j the restriction of h to V_j. Since B is a sheaf, the compositions f_ih_j for all h(V_j) Í U_i determine a g' Î hom (Z, B). We obtain a map hom (Z, A) ® hom (Z, B) given by g ® g' for each Z Î C. The collection of all these maps forms a morphism f Î hom (A, B) whose restriction to each U_i is f_i. This proves that C^~ is strict. 
Next we prove that C^~ is Cauchy-complete. Let {A_i} be a family of sheaves on C. For each i ¹ j, suppose given an open subset U_ij Í |A_i|. Suppose also given for each i ¹ j an isomorphism of sheaves u_ij: U_ij ® U_ji such that for each i, j, k these isomorphisms are compatible with each other in the sense of the definition of Cauchy-complete metric sites (2.1.3). We glue these A_i along u_ij as presheaves, and take the associated sheaf A. Then A together with the associated morphisms A_i ® A satisfies the conditions of (2.1.3). This proves that C^~ is Cauchy-complete. 

For any standard metric site C denote by G^~(C), F^~(C) and E^~(C) the Cauchy-completion, finite Cauchy-completion, and effective completion of C in C^~ respectively, then they are the Cauchy-completion, finitely Cauchy-completion, and effective completion of C respectively by (2.1.10), because C^~ is Cauchy-complete, therefore finitely Cauchy-complete and effective. Thus we obtain 

Theorem 2.3.5. Any strict metric site C has a Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion). Any Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion) of C is canonically equivalent to G^~(C) (resp. F^~(C), resp. E^~(C)) via the functor X d hom (~, X) for any X Î C'. 

Example 2.3.6. Consider the site C = W(X) of open sets of a nonempty topological space X. Denote by Sh(X) = C^~ the site of sheaves of sets on X. 
(a) Any morphism in Sh(X) is a local isomorphism and X is the final object of Sh(X). 
(b) Sh(X) is the Cauchy-completion of W(X). 
(c) For any sheaf A on X the space |A| of A is usually called the espace étalé of A, denoted Spé(A). 
(d) The metric topology Spé: Sh(X) ® Top on Sh(X) induces an equivalence from Sh(X) to the subcategory T of Top_/X consisting of local homeomorphisms to X. Thus T is also a Cauchy-completion of W(X). Note that each Spé(A) is a local space over Top_/X in the sense of (3.2).